A retooled bill providing medical care for firefighters and other emergency responders to the September 11, 2001 attacks could be resurrected soon in the Senate, a few weeks after Republicans blocked the measure, backers said on Sunday.

“We believe we are on a path to victory by the end of this week,” said Senator Charles Schumer. But he was quick to add that unexpected obstacles could arise.

He and fellow New York Senator Kirsten Gillibrand outlined for reporters some changes they will propose to their bill in an attempt to win over enough Republican support for passage as Congress winds down its legislative session for the year.

They hope to do that by producing a less expensive bill that they said would end up paying for itself, with $57 million left over in a 10-year period. That money could be used for deficit reduction, they said.

Instead of costing $7.4 billion, Gillibrand and Schumer said the measure’s price tag would be reduced to $6.2 billion.

Their revised legislation would impose a new 2 percent fee on goods and services from firms in foreign countries that are not members of the Agreement on Government Procurement. Gillibrand said Saudi Arabia would be one of the countries in that category.

Other ways the $6.2 billion cost of the health bill would be covered were by continuing a fee on travelers to the United States that is set to expire in 2015 and continuing another fee for outsourcing companies that have more than half of their employees on visas to work in the United States.

Thousands of firefighters, police and other rescue and cleanup workers have experienced respiratory problems and other illnesses from working at the World Trade Center site in the aftermath of the attack on the twin towers.

Schumer said that Senate Majority LeaderHarry Reid has promised to bring the 9/11 bill to the Senate floor for debate as soon as it wraps up work on whether to approve the New START nuclear arms control treaty with Russia.

But still unclear is whether the House of Representatives would pass the bill this year if the Senate approves it. The House already has passed its own version of the bill.